An improved order of approximation for thin-plate spline intepolation in the unit disc (Q1968787)

It is proved that for \(1\leq p \leq \infty\) the \(L_p\)-norm of the error of thin-plate spline interpolation to a \(C^\infty\) function in the unit disc decays like \(O(h^{\gamma_p+1/2})\) as \(h\to 0\) with \(\gamma_p:=\min(2,1+2/p)\) when the set \(\Xi\) of interpolation points contains the grid points \(\{hj:j\in Z^2,|hj|<1-h\}\). This improves for the special case of the disc a result of \textit{J. Duchon} [RAIRO, Anal. Numer. 12, 325-334 (1978; Zbl 0403.41003)], who proved for more general regions \(\Omega\) and grids \(\Xi\) the approximation order \(O(\delta^{\gamma_p})\) where \(\delta\) is the distance between \(\Omega\) and \(\Xi\).